From boundary random walks to Feller's Brownian Motions

TL;DR

This paper proves an invariance principle linking boundary random walks on natural numbers with Feller's Brownian motions on the half-line, encompassing various classical boundary behaviors and including boundary jumps.

Contribution

It establishes a weak convergence result connecting boundary random walks with a broad class of Feller's Brownian motions, including those with jumps at the boundary.

Findings

Weak convergence of rescaled boundary random walks to Feller's Brownian motions.

Construction of boundary random walks approximating various boundary behaviors.

Inclusion of boundary jumps governed by a measure in the Feller's Brownian motion class.

Abstract

We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$ coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple $(p_1,p_2,p_3,p_4)$. This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from $0$ governed by the measure $p_4$. For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.